Module 1.1 - Learning With Derivatives

Training Data

• Set of datapoints, each $(x,y)$

split_graph(s1, s2)

Math

• Linear Model

$$m(x; w, b) = x_1 \times w_1 + x_2 \times w_2 + b $$

def forward(self, x1: float, x2: float) -> float:
    return self.w1 * x1 + self.w2 * x2 + self.b

Model 1

• Linear Model

@dataclass
class Linear:
    # Parameters
    w1: float
    w2: float
    b: float

    def forward(self, x1: float, x2: float) -> float:
        return self.w1 * x1 + self.w2 * x2 + self.b

Decision Boundary: Model 1

model = Linear(1, 1, -0.9)
draw_graph(model)

Distance Determines Fit

• $|m(x)|$ / correct or incorrect

with_points(s1, s2, Linear(1, 1, -0.4))

Log Sigmoid Loss

def point_loss(x):
    return -math.log(minitorch.operators.sigmoid(-x))

graph(point_loss, [], [])

graph(point_loss, [], [-2, -0.2, 1])

Lecture Quiz

Outline

• Model Fit
• Symbolic Derivatives
• Numerical Derivatives
• Module 1

Model Fitting

Class Goal

• Find parameters that minimize loss

hcat([show(Linear(1, 1, -0.6)),
      show(Linear(1, 1, -0.7)),
      show(Linear(1, 1, -0.8))], 0.3)

Numerical Optimization

• Many, many different approaches
• Our focus: gradient descent
• Workhorse of modern machine learning

Iterative Parameter Fitting

1. Compute the loss function, $L(w_1, w_2, b)$
2. See how small changes would change the loss
3. Update to parameters to locally reduce the loss

Example: Update Bias

model1 = Linear(1, 1, -0.4)
model2 = Linear(1, 1, -0.5)

compare(model1, model2)

Step 1: Compute Loss

with_points(s1, s2, Linear(1, 1, -1.5))

def point_loss(out, y=1):
    return (y * # Correct Side
            -math.log(minitorch.operators.sigmoid( # Log-Sigmoid
                -out))) # Distance

Full Loss

def full_loss(m):
    l = 0
    for x, y in zip(s.X, s.y):
        l += point_loss(-m.forward(*x), y)
    return -l

hcat([graph(point_loss, [], [-2, -0.2, 1]),
      graph(lambda x: point_loss(-x), [-1, 0.4, 1.3], [])], 0.3)

Step 2: Find Direction of Improvement

hcat([show(Linear(1, 1, -1.5)),
      show(Linear(1, 1, -1.45))], 0.3)

Step 3: Update Parameters Iteratively

set_svg_height(300)
show_loss(full_loss, Linear(1, 1, 0))

Our Challenge

How do we find the right direction?

Symbolic Derivatives

Review: What is a Derivative?

How small changes in input impact output.

• $f(x)$ - function
• $x$ - point
• $f'(x)$ - "rise/run"

Review: Derivative

$$f(x) = x^2 + 1$$

def f(x):
    return x * x + 1.0

plot_function("f(x)", f)

Review: Derivative

$$f(x) = x^2 + 1$$ $$f'(x) = 2x$$

def f_prime(x):
    return 2 * x

def tangent_line(slope, x, y):
    def line(x_):
        return slope * (x_ - x) + y

    return line

plot_function("f(x) vs f'(2)",
              f, fn2=tangent_line(f_prime(2), 2, f(2)))

Symbolic Derivative

• Standard high-school derivatives
• Rewrite $f$ to new form $f'$
• Produces mathematical function

Example Function

$$f(x) = \sin(2 x)$$

plot_function("f(x) = sin(2x)", lambda x: math.sin(2 * x))

Symbolic Derivative

$$f(x) = \sin(2 x) \Rightarrow f'(x) = 2 \cos(2 x)$$

plot_function("f'(x) = 2*cos(2x)", lambda x: 2 * math.cos(2 * x),
              fn2=lambda x: math.sin(2 * x))

Multiple Arguments

$$f(x, y) = \sin(x) + \cos(y)$$

plot_function3D("f(x, y) = sin(x) + 2 * cos(y)", lambda x,y: math.sin(x) + 2 * math.cos(y))

Derivatives with Multiple Arguments

$$f_x'(x, y) = \cos(x) \ \ \ f_y'(x, y) = -2 \sin(y)$$

plot_function3D("f'_x(x, y) = cos(x)", lambda x, y: math.cos(x))

Review: Symbolic Derivatives

Expectation: Apply basic derivative rules.

• Differentiation Rules

Numerical Derivatives

What if we don't have symbols?

$$f(x) = ...$$ $$f'(x) = ...$$

For example if $f$ is unseen code.

def f(x: float) -> float:
    ...

Derivative as higher-order function

$$f(x) = ...$$ $$f'(x) = ...$$

def derivative(f: Callable[[float], float]
              ) -> Callable[[float], float]:
    def f_prime(x: float) -> float:
        ...
    return f_prime

Definition of Derivative

$$f'(x) = \lim_{\epsilon \rightarrow 0} \frac{f(x + \epsilon) - f(x - \epsilon)}{2\epsilon}$$

Central Difference

Approximate derivative

$$f'(x) \approx \frac{f(x + \epsilon) - f(x-\epsilon)}{2\epsilon}$$

Approximating Derivative

Key Idea: Only need to call $f$.

def central_difference(f: Callable[[float], float],
                       x: float) -> float:
    ...

Derivative as higher-order function

$$f(x) = ...$$ $$f'(x) = ...$$

def derivative(f: Callable[[float], float]
              ) -> Callable[[float], float]:
    def f_prime(x: float) -> float:
        return minitorch.central_difference(f, x)
    return f_prime

Advanced: Mulitiple Arguments

Turn 2-argument function into 1-arg.

def f(x, y):
    ...
    
def f_x_prime(x: float, y: float) -> float:
    def inner(x: float) -> float:
        return f(x, y)
    return derivative(inner)(x)

Example

def sigmoid(x: float) -> float:
    if x >= 0:
        return 1.0 / (1.0 + math.exp(-x))
    else:
        return math.exp(x) / (1.0 + math.exp(x))

plot_function("sigmoid", sigmoid)

Example

sigmoid_prime = derivative(sigmoid)

plot_function("Derivative of sigmoid", sigmoid_prime)

Symbolic

• Transformation of mathematical function
• Gives full form of derivative
• Utilizes mathematical identities

Numerical

• Only requires evaluating function
• Computes derivative at a point
• Can be applied to fully black-box function

Next Class: Autodifferentiation

• Computes derivative on programs trace
• Efficient for large number of parameters
• Works directly on python code

Module-1

Module-1 Learning Objectives

• Practical understanding of derivatives
• Dive into autodifferentiation
• Parameters and their usage

Module-1: What is it?

• Numerical and symbolic derivatives
• Implement our numerical class
• Implement autodifferentiation
• Everything is scalars for now (no "gradients")

Module-1 Overview

• 5 Tasks
• Module 1

Q&A